Steiner Inellipse

Author

Eric W. Weisstein
January 12, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneGeometry/SteinerInellipse.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/SteinerInellipse.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Equation

Inconic parameters

{x,y,z}={a,b,c};

Trilinear Equations

eqn=x^2 α^2+y^2β^2+z^2γ^2-2y z β γ-2z x γ α-2x y α β

Center

(1/(2a b c)){c y+b z,a z+c x,b x+a y}

CenterFunction[Centroid]

Brianchon Point

1/{x,y,z}

CenterFunction[Centroid]

Graphic

Inellipse is the circumellipse of the medial triangle

Solve[1/(a α)+1/(b β)+1/(c γ)==0/.γ->1,β]

t=Triangle[{{0,0},{1,0},{.3,.5}}];
p1=Point@TrilinearToCartesian[t,CyclicTrilinears[(((2 (a^2)-(b^2)-(c^2)))^2)/a]]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]];
p2=X[115][t];
Show[Graphics[{
t,
{
Green,
mt=MedialTriangle[t]
},
{
Blue,
Medians[t]
},
{
Red,
ParametricPlot[Evaluate[TrilinearToCartesian[mt,{α,-((a c α)/(b (c+a α))),1}/.Thread[{a,b,c}->SideLengths[Evaluate[mt]]]]],{α,-30,30},PlotRange->All,PlotPoints->50,
DisplayFunction->Identity][[1,1,1]]/.Line:>ClosedLine
},
{
Purple,
ParametricPlot[Evaluate[TrilinearToCartesian[t,{α,-((a c α)/(b (c+a α))),1}/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]]],{α,-30,30},PlotRange->All,PlotPoints->50,
DisplayFunction->Identity][[1,1,1]]/.Line:>ClosedLine
},
{
Red,
Line[Coordinates/@{p1,p2}]
},
PointLabels[Join[
Transpose[{Point/@Vertices[t],{"A","B","C"}}],
Append[#,Blue]&/@Transpose[{Midpoints[t],{"(Subscript[M, A])","(Subscript[M, B])","(Subscript[M, C])"}}],
{
{Centroid[t],"G",Blue},
{X[115][t],"(Subscript[X, 115])",{Red,LabelOffset->{-.07,-.02}}},
{p1,"(Subscript[X, 2482])",{Red}},
{X[1015][t],"(Subscript[X, 1015])",{Red,LabelOffset->{-.09,.01}}},
{X[1084][t],"(Subscript[X, 1084])",{Red,LabelOffset->{-.09,.04}}},
{X[1086][t],"(Subscript[X, 1086])",{Red,LabelOffset->{-.09,0}}},
{X[1146][t],"(Subscript[X, 1146])",{Red,LabelOffset->{-.09,0}}}
}
],
{PointSize[.025],LabelOffset->.04}
]
}],
PlotRange->All,AspectRatio->Automatic]

Incident Centers

(ctrs=IncidentCenters[KimberlingCenters,SteinerInellipse])//Timing

First/@ctrs

Length[ctrs]

{#,CenterName[#]}&/@ctrs//TableForm

Check

Short/@(ExpandAll[DistanceTrilinears[TrigExpand[eqn/.Thread[{α,β,γ}->Trilinears[#]]]],
TriangleQ[]]&/@ctrs)//Timing